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Class 10·Math
Class 10 · Math

Quadratic Equations

Quadratic equations appear everywhere—from calculating the trajectory of a thrown ball to designing arches in buildings to optimizing business profits.

Feynman Lens

Start with the simplest version: this lesson is about Quadratic Equations. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.

Quadratic equations appear everywhere—from calculating the trajectory of a thrown ball to designing arches in buildings to optimizing business profits. This chapter teaches you how to solve equations where the variable is squared (like x² terms). You'll learn multiple methods to find solutions and discover how the "discriminant" tells you in advance whether solutions exist. Quadratic equations are the bridge between linear algebra and more advanced mathematics, representing the first time many relationships become "curved" rather than straight.

What Is a Quadratic Equation?

A quadratic equation has the standard form: ax² + bx + c = 0, where a ≠ 0.

Why the restriction a ≠ 0? If a were zero, you'd have bx + c = 0, which is linear (as in chapter-03-pair-of-linear-equations), not quadratic.

Real-world example: A building wants a prayer hall with:

This is a quadratic equation that determines the hall's dimensions.

Three Methods to Solve Quadratic Equations

1. Factorization Method: Breaking Into Linear Pieces

If you can express ax² + bx + c = a(x − p)(x − q), then solutions are x = p and x = q.

Example:

Connection to chapter-02-polynomials: The Factor Theorem tells us that if (x − a) is a factor, then a is a zero of the polynomial.

2. Completing the Square Method: Algebraic Transformation

Rewrite ax² + bx + c as a perfect square plus/minus a constant.

Process:

Intuition: This method reveals the vertex of the parabola, which is the "turning point" of the quadratic.

3. Quadratic Formula: Universal Solution

For any quadratic ax² + bx + c = 0:

x = [−b ± √(b² − 4ac)] / 2a

The discriminant Δ = b² − 4ac tells the story:

Example: For 2x² + 3x + 1 = 0:

The Nature of Roots: Predicted by the Discriminant

Before solving, the discriminant tells you what type of roots to expect:

This predictive power is enormously useful in applications where you need to know if solutions exist before investing time in calculations.

Relationship Between Roots and Coefficients

If α and β are the two roots of ax² + bx + c = 0:

These relationships allow you to form a quadratic equation if you know its roots, or to check your solutions quickly.

Quadratic equations extend into:

Key Formulas and Theorems

Socratic Questions

  1. If the discriminant of a quadratic equation is negative, why can't you find a real solution using the quadratic formula? What does this mean geometrically?
  1. A quadratic equation has roots 3 and 5. Without expanding, can you write down the equation? How did you do it?
  1. If you're designing a rectangular garden with fixed perimeter, why is area optimization a quadratic problem?
  1. Why might the "completing the square" method be considered more elegant than factorization, even though it takes more steps?
  1. In real-world problems like projectile motion, what does a negative discriminant tell you about the situation being modeled?
🃏 Flashcards — Quick Recall
Term / Concept
What is Quadratic Equations?
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Quadratic Equations is the central idea of this lesson. Use the chapter examples to explain what it means and why it matters.
Term / Concept
What is Why the restriction a ≠ 0??
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If a were zero, you'd have bx + c = 0, which is linear (as in chapter-03-pair-of-linear-equations), not quadratic.
Term / Concept
What is Real-world example?
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A building wants a prayer hall with:
Term / Concept
What is Connection to chapter-02-polynomials?
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The Factor Theorem tells us that if (x − a) is a factor, then a is a zero of the polynomial.
Term / Concept
What is Process?
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- Divide by a to get x² + (b/a)x + c/a
Term / Concept
What is Intuition?
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This method reveals the vertex of the parabola, which is the "turning point" of the quadratic.
Term / Concept
What is The discriminant Δ = b² − 4ac tells the story?
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- If Δ > 0: Two distinct real solutions (parabola crosses x-axis twice)
Term / Concept
What is Example?
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For 2x² + 3x + 1 = 0:
Term / Concept
What is Real and distinct roots?
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(Δ > 0): Two different x-values satisfy the equation
Term / Concept
What is Real and equal roots?
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(Δ = 0): The parabola just touches the x-axis
Term / Concept
What is Complex roots?
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(Δ < 0): Solutions involve imaginary numbers; no real intersection with x-axis
Term / Concept
What is Standard form?
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ax² + bx + c = 0, a ≠ 0
Term / Concept
What is Quadratic formula?
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x = [−b ± √(b² − 4ac)] / 2a
Term / Concept
What is If roots are known?
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Equation is a[x² − (α + β)x + αβ] = 0
Term / Concept
What is the core idea of What Is a Quadratic Equation??
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A quadratic equation has the standard form: ax² + bx + c = 0, where a ≠ 0. Why the restriction a ≠ 0? If a were zero, you'd have bx + c = 0, which is linear (as in chapter-03-pair-of-linear-equations), not quadratic.
Term / Concept
What is the core idea of 1. Factorization Method: Breaking Into Linear Pieces?
tap to flip
If you can express ax² + bx + c = a(x − p)(x − q), then solutions are x = p and x = q.
Term / Concept
What is the core idea of 2. Completing the Square Method: Algebraic Transformation?
tap to flip
Rewrite ax² + bx + c as a perfect square plus/minus a constant.
Term / Concept
What is the core idea of 3. Quadratic Formula: Universal Solution?
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For any quadratic ax² + bx + c = 0: x = [−b ± √(b² − 4ac)] / 2a The discriminant Δ = b² − 4ac tells the story: - If Δ > 0: Two distinct real solutions (parabola crosses x-axis twice) - If Δ = 0: One repeated real…
Term / Concept
What is the core idea of The Nature of Roots: Predicted by the Discriminant?
tap to flip
Before solving, the discriminant tells you what type of roots to expect: - Real and distinct roots (Δ > 0): Two different x-values satisfy the equation - Real and equal roots (Δ = 0): The parabola just touches the…
Term / Concept
What is the core idea of Relationship Between Roots and Coefficients?
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If α and β are the two roots of ax² + bx + c = 0: - Sum of roots: α + β = −b/a - Product of roots: αβ = c/a These relationships allow you to form a quadratic equation if you know its roots, or to check your solutions…
Term / Concept
What is the core idea of Connecting to Related Topics?
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Quadratic equations extend into: - chapter-02-polynomials: Quadratics are second-degree polynomials with specific properties - chapter-05-arithmetic-progressions: Some progression problems lead to quadratic equations -…
Term / Concept
What is the core idea of Key Formulas and Theorems?
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- Standard form: ax² + bx + c = 0, a ≠ 0 - Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a - Discriminant: Δ = b² − 4ac - Sum of roots: −b/a - Product of roots: c/a - If roots are known: Equation is a[x² − (α + β)x +…
Term / Concept
What is Carpet area of 300 square meters?
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Carpet area of 300 square meters
Term / Concept
What is Length one meter more than twice the?
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Length one meter more than twice the breadth
Term / Concept
What is If breadth = x meters, then length?
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If breadth = x meters, then length = (2x + 1) meters
Term / Concept
What is Area?
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x(2x + 1) = 300 → 2x² + x − 300 = 0
Term / Concept
What is Factor?
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(x − 2)(x − 3) = 0
Term / Concept
What is Solutions?
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x = 2 or x = 3
Term / Concept
What is Divide by a to get x² +?
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Divide by a to get x² + (b/a)x + c/a
Term / Concept
What is Add and subtract [(b/2a)²] to create a?
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Add and subtract [(b/2a)²] to create a perfect square
Term / Concept
What is Solve the resulting simpler equation?
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Solve the resulting simpler equation
Term / Concept
What is If Δ > 0?
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Two distinct real solutions (parabola crosses x-axis twice)
Term / Concept
What is x = −1/2 or x = −1?
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x = −1/2 or x = −1
Term / Concept
What is Real and distinct roots (Δ > 0)?
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Two different x-values satisfy the equation
Term / Concept
What is Real and equal roots (Δ = 0)?
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The parabola just touches the x-axis
Term / Concept
What is Complex roots (Δ < 0)?
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Solutions involve imaginary numbers; no real intersection with x-axis
Term / Concept
What is Sum of roots?
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α + β = −b/a
Term / Concept
What is chapter-02-polynomials?
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Quadratics are second-degree polynomials with specific properties
Term / Concept
What is chapter-05-arithmetic-progressions?
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Some progression problems lead to quadratic equations
Term / Concept
What is chapter-06-triangles?
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Finding dimensions and areas often involves quadratic equations
40 cards — click any card to flip
📝 Quick Quiz — Test Yourself
If the discriminant of a quadratic equation is negative, why can't you find a real solution using the quadratic formula? What does this mean geometrically?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
A quadratic equation has roots 3 and 5. Without expanding, can you write down the equation? How did you do it?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
If you're designing a rectangular garden with fixed perimeter, why is area optimization a quadratic problem?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
Why might the "completing the square" method be considered more elegant than factorization, even though it takes more steps?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
In real-world problems like projectile motion, what does a negative discriminant tell you about the situation being modeled?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
Which approach best shows that you understand Quadratic Equations?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Why the restriction a ≠ 0??
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real-world example?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Connection to chapter-02-polynomials?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Process?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Intuition?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The discriminant Δ = b² − 4ac tells the story?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Example?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real and distinct roots?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real and equal roots?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Complex roots?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Standard form?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Quadratic formula?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand If roots are known?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand What Is a Quadratic Equation??
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand 1. Factorization Method: Breaking Into Linear Pieces?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand 2. Completing the Square Method: Algebraic Transformation?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand 3. Quadratic Formula: Universal Solution?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The Nature of Roots: Predicted by the Discriminant?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Relationship Between Roots and Coefficients?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Connecting to Related Topics?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Key Formulas and Theorems?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Carpet area of 300 square meters?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Length one meter more than twice the?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand If breadth = x meters, then length?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Area?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Factor?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Solutions?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Divide by a to get x² +?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Add and subtract [(b/2a)²] to create a?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Solve the resulting simpler equation?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand If Δ > 0?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand x = −1/2 or x = −1?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real and distinct roots (Δ > 0)?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real and equal roots (Δ = 0)?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
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